find the range of x.

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\(\large \color{black}{\begin{align}& \{a,b\}\in \mathbb{R^{>0}},\ \ a>b,\ \ \normalsize \text{then }\ \ a^{1/x}>b^{1/x} \hspace{.33em}\\~\\ & \normalsize \text{find the range of }\ x .\hspace{.33em}\\~\\ \end{align}}\)
i think x>0
example http://www.wolframalpha.com/input/?i=solve+87%5E%281%2Fx%29%3E78%5E%281%2Fx%29

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\[a^{1/x}\gt b^{1/x}\] divide \(b^{1/x}\) both sides and get \[\left(\frac{a}{b}\right)^{1/x}\gt 1\] taking log both sides \[x*\ln\left(\frac{a}{b}\right) \gt 0\] since \(\ln\left(\frac{a}{b}\right) \gt 0\) for \(a\gt b\), dividing it both sides wont flip the signs : \[x\gt 0\]
but after taking \(\ln \) on both sides it should be this ? \(\dfrac1x \times \ln\left(\dfrac{a}{b}\right) \gt 0\)
Ahh sry, it was just a typo
fixed : \[a^{1/x}\gt b^{1/x}\] divide \(b^{1/x}\) both sides and get \[\left(\frac{a}{b}\right)^{1/x}\gt 1\] taking log both sides \[\frac{1}{x}*\ln\left(\frac{a}{b}\right) \gt 0\] since \(\ln\left(\frac{a}{b}\right) \gt 0\) for \(a\gt b\), dividing it both sides wont flip the signs : \[\frac{1}{x}\gt 0\] Multipilying \(x^2\) both sides gives \[x\gt 0\]

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